全循環質數

在數論中,全循環質數[1]:166又名長質數是指一個質數p,使分數1/p的循環節長度比質數少1,更精確地說,全循環質數是指一個質數p,在一個已知底數為b的進位制下,在下面算式中可以得出一個循環數的質數

p為23,b為17,所得的數字0C9A5F8ED52G476B1823BE為循環數

0C9A5F8ED52G476B1823BE × 1 = 0C9A5F8ED52G476B1823BE
0C9A5F8ED52G476B1823BE × 2 = 1823BE0C9A5F8ED52G476B
0C9A5F8ED52G476B1823BE × 3 = 23BE0C9A5F8ED52G476B18
0C9A5F8ED52G476B1823BE × 4 = 2G476B1823BE0C9A5F8ED5
0C9A5F8ED52G476B1823BE × 5 = 3BE0C9A5F8ED52G476B182
0C9A5F8ED52G476B1823BE × 6 = 476B1823BE0C9A5F8ED52G
0C9A5F8ED52G476B1823BE × 7 = 52G476B1823BE0C9A5F8ED
0C9A5F8ED52G476B1823BE × 8 = 5F8ED52G476B1823BE0C9A
0C9A5F8ED52G476B1823BE × 9 = 6B1823BE0C9A5F8ED52G47
0C9A5F8ED52G476B1823BE × A = 76B1823BE0C9A5F8ED52G4
0C9A5F8ED52G476B1823BE × B = 823BE0C9A5F8ED52G476B1
0C9A5F8ED52G476B1823BE × C = 8ED52G476B1823BE0C9A5F
0C9A5F8ED52G476B1823BE × D = 9A5F8ED52G476B1823BE0C
0C9A5F8ED52G476B1823BE × E = A5F8ED52G476B1823BE0C9
0C9A5F8ED52G476B1823BE × F = B1823BE0C9A5F8ED52G476
0C9A5F8ED52G476B1823BE × G = BE0C9A5F8ED52G476B1823
0C9A5F8ED52G476B1823BE × 10 = C9A5F8ED52G476B1823BE0
0C9A5F8ED52G476B1823BE × 11 = D52G476B1823BE0C9A5F8E
0C9A5F8ED52G476B1823BE × 12 = E0C9A5F8ED52G476B1823B
0C9A5F8ED52G476B1823BE × 13 = ED52G476B1823BE0C9A5F8
0C9A5F8ED52G476B1823BE × 14 = F8ED52G476B1823BE0C9A5
0C9A5F8ED52G476B1823BE × 15 = G476B1823BE0C9A5F8ED52

,循環節長度為22,比23少1,因此23為全循環質數


十進位中的全循環質數有:

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593,... OEIS數列A001913

參見

參考文獻

  1. Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
  1. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.
  2. Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers"; in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240–246.

外部連結

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